I saw some very similar questions but for windows and I am using unity and have no space to install a game so I need to use a USB drive but have no idea how to unlock the /media file to read/write so I can write to it as well.
I know from using Steam that you can "add a library location" in your downloads tab. After that, when you download some game, you'll be presented with the option for where to install it.
If you wish to move your already installed games, this Steam help document should help you do that.
You'll have to make sure that your are technically the "owner" of the USB drive so Steam has permissions to access.
If you're having trouble mounting something or a filesystem error, this post should help you get an external partition mounted for use by Steam. If you're still not getting that, try this article, mentioning using games already installed on a Windows partition.